Double-diffusive transport in multicomponent vertical convection

Motivated by the ablation of vertical ice faces in salt water, we use three-dimensional direct numerical simulations to investigate the heat and salt fluxes in two-scalar vertical convection. For parameters relevant to ice-ocean interfaces in the convection-dominated regime, we observe that the salinity field drives the convection and that heat is essentially transported as a passive scalar. By varying the diffusivity ratio of heat and salt (i.e., the Lewis number Le), we identify how the different molecular diffusivities affect the scalar fluxes through the system. Away from the walls, we find that the heat transport is determined by a turbulent Prandtl number of ${\mathrm{Pr}}_t\approx 1$ and that double-diffusive effects are practically negligible. However, the difference in molecular diffusivities plays an important role close to the boundaries. In the (unrealistic) case where salt diffused faster than heat, the ratio of salt-to-heat fluxes would scale as ${\mathrm{Le}}^{1/3}$, consistent with classical nested scalar boundary layers. However, in the realistic case of faster heat diffusion (relative to salt), we observe a transition towards a ${\mathrm{Le}}^{1/2}$ scaling of the ratio of the fluxes. This coincides with the thermal boundary layer width growing beyond the thickness of the viscous boundary layer. We find that this transition is not determined by a critical Lewis number, but rather by a critical Prandtl number $\mathrm{Pr}\approx 10$, slightly below that for cold seawater where $\mathrm{Pr}=14$. We compare our results to similar studies of sheared and double-diffusive flow under ice shelves, and discuss the implications for fluxes in large-scale ice-ocean models. By coupling our results to ice-ocean interface thermodynamics, we describe how the flux ratio impacts the interfacial salinity, and hence the strength of solutal convection and the ablation rate.

Figure 1

(a) A schematic of the simulation domain, featuring two no-slip vertical planes. (b, c) Volume renderings of the instantaneous temperature and salinity fields adjacent to the wall at $x=0$ from simulations (b) A10L10 and (c) A100L100. Colorbars show the opacity used for the volume rendering as well as the color, and the bounding box outlines the full extent of the domain. Slender, green plume structures highlight the buoyant regions of low salinity driving the flow up the wall, surrounded by more diffuse blue regions highlighting the low temperature patches. The buoyant flow due to the salinity perturbations advects these cold patches, so the structures in the two scalar fields become strongly correlated. With a higher $\mathrm{Sc}$ and Le, the green salinity structures in (c) are thinner and are nested more deeply in the diffuse temperature structures compared to those observed in (b).

Figure 2

Wall-normal profiles of flow quantities from simulations P10 (red dashed), A10L10 (pink), P100 (blue dashed), and A100L100 (cyan). Solid lines show the time-averaged profiles and the shaded regions highlight the standard deviation (in time) of the $yz$-averaged profiles. Quantities plotted are (a) mean concentration, (b) concentration variance, (c) mean temperature, (d) temperature variance, (e) mean vertical velocity, and (f) wall-normal velocity variance.

Figure 3

Instantaneous plane snapshots from simulation A100L100, where $\mathrm{Sc}=100$ and $\mathrm{Le}=100$. In-plane velocity vectors are overlaid on each snapshot, which for the wall panels (b) and (c) correspond to the local shear stress at the wall. (a) Concentration field in the $xz$ plane $y=2H$. (b, c) Wall-normal dimensionless flux of concentration and temperature at $x=0$ as defined in (12). (d, e) Concentration and temperature fields at the center $yz$ plane $x=H/2$. (f) Temperature field in the $xz$ plane $y=2H$. (g, h) Concentration and temperature fields in the $xy$ plane $z=4H$. Since the concentration field drives the flow, regions of high or low concentration coincide with stronger vertical velocities. Structures in the temperature field are far more diffuse than those in the concentration field due to the large Lewis number.

Figure 4

Ratio $R$ of solutal Nusselt number ${\mathrm{Nu}}_C$ to thermal Nusselt number ${\mathrm{Nu}}_T$ across all the simulations plotted against the Lewis number. Colors denote the Schmidt number of the simulations, and crosses are used to highlight the cases with zero density ratio. The dashed straight lines with slopes $1/3$ and $1/2$ are shown for comparison. The black dot highlights the fixed theoretical point $R=\mathrm{Le}=1$.

Figure 5

Thermal (circles) and solutal (squares) boundary layer widths in viscous units plotted against (a) the Lewis number and (b) the Prandtl or Schmidt number. The viscous sublayer, defined as $x^{+}\lesssim 5$, is highlighted by the red shading. Data points for $\mathrm{Le}=1$ in (a) are inferred from the solutal boundary layer widths for each dataset (A10 and A100). For $\mathrm{Le}=1$ the thermal and solutal boundary layer widths must be equal.

Figure 6

Wall-normal profiles of the turbulent heat diffusivity $K_T$ for each simulation, for two different Schmidt numbers, $\text{(a)}\mathrm{Sc}=10$ and $\text{(b)}\mathrm{Sc}=100$. Curves labeled as $\mathrm{Le}=1$ represent the turbulent diffusivity of the concentration field. For comparison, the wall-normal profile of the turbulent viscosity ${\nu }_t$ is also plotted as a dark dashed line, and the constant values of molecular diffusivity ${\kappa }_T$ are plotted as dotted lines with colors matching the legend. Shaded regions highlight the temporal standard deviation of the turbulent heat flux $\overline{u^{\prime }{T}^{\prime }}$, normalized by the mean temperature gradient ${\partial }_x\overline{T}$ (multiplied by ${\kappa }_T$).

Figure 7

Data from [28] showing the width of the diffusive solutal sublayer in viscous units ${\delta }_C^{+}={\delta }_C{V}^{*}/\nu$. The viscous sublayer $x^{+}\lesssim 5$ is shaded in the same manner as in Fig. 5. The solutal sublayer width depends primarily on $\mathrm{Sc}$, with a very slight increase observed with $\mathrm{Ra}$.

Figure 8

Dependence of melt rate and interface salinity $C_i$ on flux ratio $R$. The ambient ocean salinity is assumed to be fixed, so a lower $C_i$ leads to a stronger buoyancy source of fresher water at the ice face, driving stronger convection and enhancing the melt rate. Vertical dashed lines mark the two values of the flux ratio assumed by Jenkins [20] (${\mathrm{Le}}^{1/3}$) and Kerr and McConnochie [15] (${\mathrm{Le}}^{1/2}$). This result assumes (i) salt flux is determined by $\mathrm{Nu}\approx 0.1{\mathrm{Ra}}^{1/3}$, (ii) ambient ocean temperature of $1{}^{\circ }\mathrm{C}$, and (iii) physical constants prescribed as in Table 2.

Figure 9

Wall-normal profiles of the scalar variance budget terms from (A2) for three simulations: (a) A10L01, (b, c) A100L10, and (d) A100L100. In (a), (c), and (d) the temperature variance budget is presented. In (b) the budget terms plotted are actually those of the concentration variance to provide insight into the dynamics at $\mathrm{Le}=1$, in which case temperature statistics would evolve identically to the concentration statistics shown in panel (b).